Almost controllable graphs and beyond

Abstract

An eigenvalue λ of a graph G of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector jn. In 1978, Cvetković proved that G has exactly one main eigenvalue if and only if G is regular, and posed the following long-standing problem: characterize the graphs with exactly k(2≤k≤n) main eigenvalues. Graphs of order n with n, n−1 main eigenvalues are called controllable, almost controllable, respectively. Cographs, threshold graphs are frequently studied in structural graph theory and computer science. In this paper, all almost controllable cographs, all almost controllable threshold graphs and all almost controllable graphs with second largest eigenvalue less than or equal to 5−12 are characterized. Furthermore, we give some results about cographs with exactly n−2 main eigenvalues, and propose some additional problems for further study.